Remarks on Stability of the Linear Functional Equation in Single Variable

2012 ◽  
pp. 91-119 ◽  
Author(s):  
Janusz Brzdȩk ◽  
Dorian Popa ◽  
Bing Xu
Keyword(s):  
Functional Equation ◽  
Linear Functional ◽  
Linear Functional Equation ◽  
Single Variable

scholarly journals Banach Limit in the Stability Problem of a Linear Functional Equation

2021 ◽  
Vol 76 (1) ◽  
Author(s):  
Roman Badora ◽  
Janusz Brzdęk
Keyword(s):  
Functional Equation ◽  
Stability Problem ◽  
Linear Functional ◽  
Linear Functional Equation ◽  
Single Variable ◽  
Banach Limit ◽  
The Stability

AbstractWe present some applications of the Banach limit in the study of the stability of the linear functional equation in a single variable.

scholarly journals On Ulam’s Type Stability of the Linear Equation and Related Issues

2014 ◽  
Vol 2014 ◽  
pp. 1-14 ◽  
Author(s):  
Janusz Brzdęk ◽  
Krzysztof Ciepliński ◽  
Zbigniew Leśniak
Keyword(s):  
Functional Equation ◽  
Linear Equation ◽  
Linear Functional ◽  
Linear Functional Equation ◽  
Single Variable ◽  
Survey Paper ◽  
Ulam’S Type Stability ◽  
Stability Results

This is a survey paper concerning stability results for the linear functional equation in single variable. We discuss issues that have not been considered or have been treated only briefly in other surveys concerning stability of the equation. In this way, we complement those surveys.

scholarly journals Weighted space method for the stability of some nonlinear equations

2012 ◽  
Vol 6 (1) ◽  
pp. 126-139 ◽  
Author(s):  
Liviu Cǎdariu ◽  
Laura Gǎvruţa ◽  
Paşc Gǎvruţa
Keyword(s):  
Functional Equation ◽  
Integral Equation ◽  
Nonlinear Equations ◽  
Volterra Integral Equation ◽  
Linear Functional ◽  
Weighted Space ◽  
Linear Functional Equation ◽  
Single Variable ◽  
The Stability ◽  
Space Method

We prove the stability of some equations in a single variable, including a non-linear functional equation, a linear functional equation as well as a Volterra integral equation, by using the weighted space method. Our results generalize and extend some recent theorems given in this field, with simplified proofs. Several direct applications of these results are also obtained.

scholarly journals Applications of Banach Limit in Ulam Stability

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 841
Author(s):  
Roman Badora ◽  
Janusz Brzdęk ◽  
Krzysztof Ciepliński
Keyword(s):  
Functional Equation ◽  
Functional Equations ◽  
Linear Functional ◽  
Linear Functional Equation ◽  
Ulam Stability ◽  
Single Variable ◽  
Cauchy Equation ◽  
Banach Limit

We show how to get new results on Ulam stability of some functional equations using the Banach limit. We do this with the examples of the linear functional equation in single variable and the Cauchy equation.

On a linear functional equation

1989 ◽  
Vol 38 (2-3) ◽  
pp. 113-122 ◽  
Author(s):  
László Székelyhidi
Keyword(s):  
Functional Equation ◽  
Linear Functional ◽  
Linear Functional Equation

scholarly journals The Generalized Pomeron Functional Equation

2019 ◽  
Vol 2019 ◽  
pp. 1-4
Author(s):  
Yong-Guo Shi
Keyword(s):  
Functional Equation ◽  
Continuous Function ◽  
Linear Functional ◽  
Continuous Solution ◽  
Suitable Condition ◽  
Single Parameter ◽  
Linear Functional Equation ◽  
Continuous Solutions ◽  
Constant Coefficients

This paper investigates the linear functional equation with constant coefficients φt=κφλt+ft, where both κ>0 and 1>λ>0 are constants, f is a given continuous function on ℝ, and φ:ℝ⟶ℝ is unknown. We present all continuous solutions of this functional equation. We show that (i) if κ>1, then the equation has infinite many continuous solutions, which depends on arbitrary functions; (ii) if 0<κ<1, then the equation has a unique continuous solution; and (iii) if κ=1, then the equation has a continuous solution depending on a single parameter φ0 under a suitable condition on f.

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